Realizing the Willis equations with pre-stresses

TL;DR

This paper demonstrates that the Willis equations can accurately model the linear elastic behavior of materials with inhomogeneous pre-stresses, incorporating material length scales and aligning with transformation elastodynamics principles.

Contribution

It proves that pre-stressed inhomogeneous materials are described by Willis equations, extending classical elasticity to include pre-stress gradients and material length scales.

Findings

Willis equations describe pre-stressed inhomogeneous materials.

Additional terms relate to pre-stress gradients.

Framework aligns with transformation elastodynamics principles.

Abstract

This paper proves that the linear elastic behavior of the material with inhomogeneous pre-stresses can be described by the Willis equations. In this case, the additional terms in the Willis equations, compared with the classical linear elastic equations for homogeneous media, are related to the gradient of pre-stresses. In this way, the material length scale is naturally incorporated in the framework of continuum mechanics. All these findings also coincide with the results of transformation elastodynamics, so that they can meet the requirement of the principle of material objectivity and the principle of general invariance.